Arithmetic

Arithmetic(ぽ う) There is an use of the following two kinds mainly.

  1. Computer scienceIn fieldAlgorithmIt is used as meaning language.
  2. Mathematics"Operation" and being synonymous in field, it is used. You detail with this manuscript concerning this.

N Section arithmetic(N like this the ぽ う) With, in a broad sense, gathering A Direct product AN Subclass D Empty A To mapping F Thing is called,D You call the domain of this arithmetic.N OptionThe number of ordersSo it is good. F It calls the number of sections. AN I < N The number of orders which are filled up I It makes the index A Original family (AI)I<N Gathering which consists of everything is displayed. A With family of the arithmetic in there R With uniting (A R)Algebra systemWith you say. Table of contents

1

Entire area arithmetic

Usually,

D

= AN There to be many times when of the case where is thought, such arithmetic F It calls entire area arithmetic () temporarily. N There is many a thing which of the case of the limited thinks number of orders. AN A N Uniting the origin (A0 A1 ..., AN-1) Being the whole,F As for those which moved this due to F(A0 A1 ..., AN-1) With it can write. N = 2 There to be many times when of the case where is thought, in this case, F(A0 A1) A0 F A1 A0 A1 F There is many a thing which is written. Therefore, entire area binary arithmetic,A Original two uniting (A

B) In each A It is of mapping which makes a some origin correspond. For example, two real numbers A B The harmony

A + B As for mapping which it makes correspond, being the entire area binary arithmetic in real number all gathering, as for sign + of harmony this arithmetic (namely addition) the above-mentioned second notation A0 F A1 F Those which hit it is undone. Medium position notationSo A + B With you write, butReverse polish notationSo A b + You write. A0 A1 F With those which hit to the notation which is said it is undone. Either 1 section arithmetic is not unusual. K Vector space above V Putting, K Optional origin A With

V Optional origin V Vis-a-vis V Origin Av Exists, but as for this,K It makes index gathering V 1 section arithmetic family (FA)A∈K It being FAV Av When so you have displayed, it is undone. Ring R OnAdditive group M In R Origin M To it seems like action, "external arithmetic" is regarded 1 section arithmetic entirely. Non entire area arithmetic Non entire area (or local) either arithmetic is not unusual. M When so displaying,M Duality A B The harmony A + B

As for mapping which it makes correspond

M

It is the binary arithmetic in, but

A B Only at the time of the same size A + B Because is not defined, it is non entire area arithmetic. A B The product AB Mapping which it makes correspond M It is the binary arithmetic in, butA Quantity in line B When the quantity of line is equal only, AB Because is not defined, after all it is non entire area arithmetic. Formal languageAs for the arithmetic in, those of non entire area are general. Either the non entire area arithmetic where the number of sections is more than 2 is not unusual. N Variable functor and N As for the variable predicate symbol, only vis-a-vis the section applicable N When section arithmetic is displayed, it is undone. The arithmetic which has the number of transfinite sections There is also an arithmetic which designates the number of transfinite orders as the number of sections. R When so you display, direct product R

ω It is actual

Series A0 A1

... It is the whole, but in the real number line which is focused that

Extremity

Mapping which it makes correspond is the ω section arithmetic of non entire area. N

With the algebra condition of ω section arithmetic rewriting type definition, it is possible to axiomatize the extremity. R The next six conditions in are satisfied, ω section arithmetic L Is not anything less than the extremity. L(A A ...) = A L(A1 A2

...) =
  1. A L(B1 B2
  2. ...) = B AN ? BN (N=1,2...If) A ? B L(A1 A2 ...) = A If A1 A2 ... Optional substring B
  3. 1 B2 ... Vis-a-vis L(B1 B2 ...) = A (It puts, principle of shooting)L(A1 A2 ...) = L(B1 B2 ...) = A
  4. AN ? CN ? BN (N=1,2...If) L(C1 C2 ...) = A (Archimedes's principle)L(A± (1/1), A± (1/2), A± (1/3), ...) = A (Compound same order) A1 A
  5. 2 ... Optional substring B1 B2 ... L(C1
  6. C2 ...) = A The substring which becomes C1 C2 ... If it is L(A1 A2 ...) = A University 1, That in the student and the high school student of 2nd year "arithmetic of queue is non entire area arithmetic", that "the extremity is ω section arithmetic", what you teach is not recommended, but for the mathematical scientist to recognize such, it probably will be desirable it meaning that range of vision spreads. Expansion of arithmetic concept As defined in beginning,F Gathers A In N It is section arithmetic with,F AN The subclass which is D
Empty

A

To it means that it is mapping. F

A

N×A Subclass {((X1 ... , XN) Y) | (X1 ... , XN) ∈D F(X1 ... , XN) = Y } With it is classed. F, G, ... N If section arithmetic, it can think of harmony as those gathering. F If it moves, in consequence of that N It moves. AN (N = 0, 1, 2, ... < Direct sum ω) as gathering A* So you display,A* × A Subclass R We have decided to call arithmetic even to thing. Wide sense Concerning arithmetic, the α ∈ A* With Y ∈ A With (α, Y) ∈ R Filling up like usual arithmetic R(α) = Y With not to be able express, α R Y With you must write. A* Vis-a-vis (α, Y) ∈ R It fills up Y Because does not limit the only. Wide sense Perhaps) it should call relationship. As for such broad arithmetic relationship,Symbolic logicIt appears often. A Putting, meaning mapping F * With under inf {F *(A1) ... , F *(AN)}? F *(B) The Boolean expression which is filled up A

1 ... , AN B (N It is optional) from it can be made A*×A Origin ((A1 ... , AN) b) の全体を R で表せば、これは上記の意味での広義の算法・関係である。 このように算法の概念と関係の概念をともに拡張して統合すると、算法と関係とを統一的に扱うことができて極めて有効である。 命名について 算法を「演算」とよぶことも多い。 しかし、ここで考えた「算法の概念」の名前としては、「算法」も「演算」も相応しくはないであろう。 「算法」は「計算の規則」あるいは「計算の方法」を連想させ、「演算」は「計算を演ずるという行為」を連想させ、ともに「写像」としての「算法の概念」を連想させにくい。 一つの写像に対しても、それの「計算の方法」は一般に複数あり、「計算する方法」も「計算の規則」も具体的に記述できない場合さえあり、人や機械が「計算を演ずるという行為」はもちろん写像とは異なるからである。 「写像」という概念が未発達で「計算」と「計算の仕方」の違いも曖昧であった時代に生まれた「算法」とか「演算」とかの言葉を用いるのが間違っているのであろう。 冒頭に記したとおり、計算機科学の分野ではアルゴリズムを「算法」とよぶこともあり、その場合には上記の意味での「算法」は「演算」とよぶ方がよいかもしれない。 しかし数学の分野では、上記の意味での「算法」という術語も昔から定着している。 むしろアルゴリズムを「算法」ではなく「計算手順」とする方が、意味からいっても先例を尊ぶ点でも、好ましいであろう。 関連項目 代数系 代数学 二項演算

 

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